How many different rearrangements are there using the letters AABBBCCCC
To find the number of different rearrangements of the letters AABBBCCCC, we can use the concept of permutations.
In this case, we have 10 characters in total: 2 A's, 3 B's, and 4 C's. We can use permutations to find the number of possible arrangements.
First, we calculate the total number of possible arrangements without any restrictions. We do this by using the formula for permutations:
P(total) = n!
where n is the total number of items.
In this case, n = 10, so P(total) = 10!.
But since we have repeated characters, we need to account for the duplicate arrangements.
1. We divide by the number of arrangements of the A's, which is 2!. This accounts for the fact that the two A's can be arranged amongst themselves without creating a new arrangement.
2. We also divide by the number of arrangements of the B's, which is 3!. This accounts for the fact that the three B's can be arranged amongst themselves without creating a new arrangement.
3. Similarly, we divide by the number of arrangements of the C's, which is 4!. This accounts for the fact that the four C's can be arranged amongst themselves without creating a new arrangement.
So, the total number of different rearrangements can be calculated as:
P(total) / (P(A) * P(B) * P(C))
= 10! / (2! * 3! * 4!)
Calculating this expression will give you the number of different rearrangements of the letters AABBBCCCC.